Abstract
In the present article we discuss the classification of quantum groups whose quasi-classical limit is a given simple complex Lie algebra \({\mathfrak{g}}\). This problem is reduced to the classification of all Lie bialgebra structures on \({\mathfrak{g}(\mathbb{K})}\), where \({\mathbb{K}=\mathbb{C}((\hbar))}\). The associated classical double is of the form \({\mathfrak{g}(\mathbb{K})\otimes_{\mathbb{K}} A}\), where A is one of the following: \({\mathbb{K}[\varepsilon]}\), where \({\varepsilon^{2}=0}\), \({\mathbb{K}\oplus\mathbb{K}}\) or \({\mathbb{K}[j]}\), where \({j^{2}=\hbar}\). The first case is related to quasi-Frobenius Lie algebras. In the second and third cases we introduce a theory of Belavin–Drinfeld cohomology associated to any non-skewsymmetric r-matrix on the Belavin–Drinfeld list (Belavin and Drinfeld in Soviet Sci Rev Sect C: Math Phys Rev 4:93–165, 1984). We prove a one-to-one correspondence between gauge equivalence classes of Lie bialgebra structures on \({\mathfrak{g}(\mathbb{K})}\) and cohomology classes (in case II) and twisted cohomology classes (in case III) associated to any non-skewsymmetric r-matrix.
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Belavin A., Drinfeld V.: Triangle equations and simple Lie algebras. Soviet Sci. Rev. Sect. C: Math. Phys. Rev. 4, 93–165 (1984)
Benkart G., Zelmanov E.: Lie algebras graded by finite root systems and intersection matrix algebras. Invent. math. 126, 1–45 (1996)
Drinfeld V.G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations. (Russian) Dokl. Akad. Nauk SSSR 268, 285–287 (1983)
Drinfeld, V.G.: Quantum groups. Proceedings ICM (Berkeley 1996). AMS 1, 798–820 (1997)
Etingof P., Kazhdan D.: Quantization of Lie bialgebras I. Sel. Math. (NS) 2, 1–41 (1996)
Etingof P., Kazhdan D.: Quantization of Lie bialgebras II. Sel. Math. (NS) 4, 213–232 (1998)
Etingof P., Schiffmann O.: Lectures on Quantum Groups. International Press, Cambridge (1988)
Montaner F., Stolin A., Zelmanov E.: Classification of Lie bialgebras over current algebras. Sel. Math. (NS) 16, 935–962 (2010)
Serre J.-P.: Local Fields. Springer-Verlag, New York (1979)
Stolin A.: Some remarks on Lie bialgebra structures on simple complex Lie algebras. Commun. Algebra 27(9), 4289–4302 (1999)
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Communicated by H.-T. Yau
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Kadets, B., Karolinsky, E., Pop, I. et al. Classification of Quantum Groups and Belavin–Drinfeld Cohomologies. Commun. Math. Phys. 344, 1–24 (2016). https://doi.org/10.1007/s00220-016-2622-y
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DOI: https://doi.org/10.1007/s00220-016-2622-y